\(\int \frac {x^6}{(c+d x)^2 (a+b x^2)^{3/2}} \, dx\) [1237]

Optimal result

Integrand size = 22, antiderivative size = 250 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=-\frac {a^2 \left (2 a c d+\left (b c^2-a d^2\right ) x\right )}{b^2 \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}-\frac {2 c \sqrt {a+b x^2}}{b^2 d^3}+\frac {x \sqrt {a+b x^2}}{2 b^2 d^2}-\frac {c^6 \sqrt {a+b x^2}}{d^3 \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {3 \left (2 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 b^{5/2} d^4}+\frac {3 c^5 \left (b c^2+2 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{d^4 \left (b c^2+a d^2\right )^{5/2}} \] Output:

-a^2*(2*a*c*d+(-a*d^2+b*c^2)*x)/b^2/(a*d^2+b*c^2)^2/(b*x^2+a)^(1/2)-2*c*(b 
*x^2+a)^(1/2)/b^2/d^3+1/2*x*(b*x^2+a)^(1/2)/b^2/d^2-c^6*(b*x^2+a)^(1/2)/d^ 
3/(a*d^2+b*c^2)^2/(d*x+c)+3/2*(-a*d^2+2*b*c^2)*arctanh(b^(1/2)*x/(b*x^2+a) 
^(1/2))/b^(5/2)/d^4+3*c^5*(2*a*d^2+b*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^ 
2)^(1/2)/(b*x^2+a)^(1/2))/d^4/(a*d^2+b*c^2)^(5/2)
 

Mathematica [A] (verified)

Time = 2.08 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.31 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {\frac {d \left (b^3 c^4 x^2 \left (-6 c^2-3 c d x+d^2 x^2\right )+a^3 d^4 \left (-8 c^2-5 c d x+3 d^2 x^2\right )-a b^2 c^2 \left (6 c^4+3 c^3 d x+7 c^2 d^2 x^2+6 c d^3 x^3-2 d^4 x^4\right )+a^2 b d^2 \left (-8 c^4-8 c^3 d x-4 c^2 d^2 x^2-3 c d^3 x^3+d^4 x^4\right )\right )}{b^2 \left (b c^2+a d^2\right )^2 (c+d x) \sqrt {a+b x^2}}+\frac {12 c^5 \left (b c^2+2 a d^2\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}}+\frac {6 \left (2 b c^2-a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{-\sqrt {a}+\sqrt {a+b x^2}}\right )}{b^{5/2}}}{2 d^4} \] Input:

Integrate[x^6/((c + d*x)^2*(a + b*x^2)^(3/2)),x]
 

Output:

((d*(b^3*c^4*x^2*(-6*c^2 - 3*c*d*x + d^2*x^2) + a^3*d^4*(-8*c^2 - 5*c*d*x 
+ 3*d^2*x^2) - a*b^2*c^2*(6*c^4 + 3*c^3*d*x + 7*c^2*d^2*x^2 + 6*c*d^3*x^3 
- 2*d^4*x^4) + a^2*b*d^2*(-8*c^4 - 8*c^3*d*x - 4*c^2*d^2*x^2 - 3*c*d^3*x^3 
 + d^4*x^4)))/(b^2*(b*c^2 + a*d^2)^2*(c + d*x)*Sqrt[a + b*x^2]) + (12*c^5* 
(b*c^2 + 2*a*d^2)*ArcTan[(Sqrt[-(b*c^2) - a*d^2]*x)/(Sqrt[a]*(c + d*x) - c 
*Sqrt[a + b*x^2])])/(-(b*c^2) - a*d^2)^(5/2) + (6*(2*b*c^2 - a*d^2)*ArcTan 
h[(Sqrt[b]*x)/(-Sqrt[a] + Sqrt[a + b*x^2])])/b^(5/2))/(2*d^4)
 

Rubi [A] (verified)

Time = 2.56 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.31, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {601, 25, 2182, 2185, 2185, 27, 719, 224, 219, 488, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^6}{\left (a+b x^2\right )^{3/2} (c+d x)^2} \, dx\)

\(\Big \downarrow \) 601

\(\displaystyle -\frac {\int -\frac {-\frac {2 c d^3 x a^4}{b^2 \left (b c^2+a d^2\right )^2}+\frac {c^2 \left (b c^2-a d^2\right ) a^3}{b^2 \left (b c^2+a d^2\right )^2}-\frac {x^2 a^2}{b^2}+\frac {x^4 a}{b}}{(c+d x)^2 \sqrt {b x^2+a}}dx}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {-\frac {2 c d^3 x a^4}{b^2 \left (b c^2+a d^2\right )^2}+\frac {c^2 \left (b c^2-a d^2\right ) a^3}{b^2 \left (b c^2+a d^2\right )^2}-\frac {x^2 a^2}{b^2}+\frac {x^4 a}{b}}{(c+d x)^2 \sqrt {b x^2+a}}dx}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2182

\(\displaystyle \frac {-\frac {\int \frac {-a \left (\frac {c^2}{d}+\frac {a d}{b}\right ) x^3+a c \left (\frac {c^2}{d^2}+\frac {a}{b}\right ) x^2-a \left (\frac {c^4}{d^3}-\frac {a^2 d}{b^2}\right ) x+\frac {a^2 c \left (b^2 c^4-a b d^2 c^2+a^2 d^4\right )}{b^2 d^2 \left (b c^2+a d^2\right )}}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}-\frac {a c^6 \sqrt {a+b x^2}}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {\frac {\int \frac {\frac {3 c d \left (b^2 c^4+a^2 d^4\right ) a^2}{b \left (b c^2+a d^2\right )}+5 c d \left (b c^2+a d^2\right ) x^2 a-\left (b c^4-2 a d^2 c^2-\frac {3 a^2 d^4}{b}\right ) x a}{(c+d x) \sqrt {b x^2+a}}dx}{2 b d^3}-\frac {a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{2 b^2 d^3}}{a d^2+b c^2}-\frac {a c^6 \sqrt {a+b x^2}}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 2185

\(\displaystyle \frac {-\frac {\frac {\frac {\int \frac {3 a d^2 \left (\frac {a c d \left (b^2 c^4+a^2 d^4\right )}{b c^2+a d^2}-\left (2 b c^2-a d^2\right ) \left (b c^2+a d^2\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{b d^2}+\frac {5 a c \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b}}{2 b d^3}-\frac {a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{2 b^2 d^3}}{a d^2+b c^2}-\frac {a c^6 \sqrt {a+b x^2}}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {\frac {3 a \int \frac {\frac {a c d \left (b^2 c^4+a^2 d^4\right )}{b c^2+a d^2}-\left (2 b c^2-a d^2\right ) \left (b c^2+a d^2\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{b}+\frac {5 a c \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b}}{2 b d^3}-\frac {a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{2 b^2 d^3}}{a d^2+b c^2}-\frac {a c^6 \sqrt {a+b x^2}}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {-\frac {\frac {\frac {3 a \left (\frac {2 b^2 c^5 \left (2 a d^2+b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )}-\frac {\left (2 b c^2-a d^2\right ) \left (a d^2+b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}\right )}{b}+\frac {5 a c \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b}}{2 b d^3}-\frac {a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{2 b^2 d^3}}{a d^2+b c^2}-\frac {a c^6 \sqrt {a+b x^2}}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {-\frac {\frac {\frac {3 a \left (\frac {2 b^2 c^5 \left (2 a d^2+b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )}-\frac {\left (2 b c^2-a d^2\right ) \left (a d^2+b c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{b}+\frac {5 a c \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b}}{2 b d^3}-\frac {a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{2 b^2 d^3}}{a d^2+b c^2}-\frac {a c^6 \sqrt {a+b x^2}}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {\frac {3 a \left (\frac {2 b^2 c^5 \left (2 a d^2+b c^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d \left (a d^2+b c^2\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 b c^2-a d^2\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )}{b}+\frac {5 a c \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b}}{2 b d^3}-\frac {a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{2 b^2 d^3}}{a d^2+b c^2}-\frac {a c^6 \sqrt {a+b x^2}}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {\frac {\frac {3 a \left (-\frac {2 b^2 c^5 \left (2 a d^2+b c^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d \left (a d^2+b c^2\right )}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 b c^2-a d^2\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )}{b}+\frac {5 a c \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b}}{2 b d^3}-\frac {a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{2 b^2 d^3}}{a d^2+b c^2}-\frac {a c^6 \sqrt {a+b x^2}}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {\frac {\frac {3 a \left (-\frac {2 b^2 c^5 \left (2 a d^2+b c^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \left (a d^2+b c^2\right )^{3/2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (2 b c^2-a d^2\right ) \left (a d^2+b c^2\right )}{\sqrt {b} d}\right )}{b}+\frac {5 a c \sqrt {a+b x^2} \left (a d^2+b c^2\right )}{b}}{2 b d^3}-\frac {a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}{2 b^2 d^3}}{a d^2+b c^2}-\frac {a c^6 \sqrt {a+b x^2}}{d^3 (c+d x) \left (a d^2+b c^2\right )^2}}{a}-\frac {a^2 \left (x \left (b c^2-a d^2\right )+2 a c d\right )}{b^2 \sqrt {a+b x^2} \left (a d^2+b c^2\right )^2}\)

Input:

Int[x^6/((c + d*x)^2*(a + b*x^2)^(3/2)),x]
 

Output:

-((a^2*(2*a*c*d + (b*c^2 - a*d^2)*x))/(b^2*(b*c^2 + a*d^2)^2*Sqrt[a + b*x^ 
2])) + (-((a*c^6*Sqrt[a + b*x^2])/(d^3*(b*c^2 + a*d^2)^2*(c + d*x))) - (-1 
/2*(a*(b*c^2 + a*d^2)*(c + d*x)*Sqrt[a + b*x^2])/(b^2*d^3) + ((5*a*c*(b*c^ 
2 + a*d^2)*Sqrt[a + b*x^2])/b + (3*a*(-(((2*b*c^2 - a*d^2)*(b*c^2 + a*d^2) 
*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (2*b^2*c^5*(b*c^2 + 
2*a*d^2)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d* 
(b*c^2 + a*d^2)^(3/2))))/b)/(2*b*d^3))/(b*c^2 + a*d^2))/a
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 601

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo 
l] :> With[{Qx = PolynomialQuotient[x^m*(c + d*x)^n, a + b*x^2, x], e = Coe 
ff[PolynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 0], f = Coeff[Pol 
ynomialRemainder[x^m*(c + d*x)^n, a + b*x^2, x], x, 1]}, Simp[(a*f - b*e*x) 
*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*(p + 1))   Int[(c 
+ d*x)^n*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*(p + 1)*Qx)/(c + d*x)^n + (e* 
(2*p + 3))/(c + d*x)^n, x], x], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1] 
 && LtQ[p, -1] && ILtQ[n, 0] && NeQ[b*c^2 + a*d^2, 0]
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(647\) vs. \(2(226)=452\).

Time = 0.50 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.59

method	result	size
risch	\(-\frac {\left (-d x +4 c \right ) \sqrt {b \,x^{2}+a}}{2 b^{2} d^{3}}-\frac {\frac {d^{3} a^{2} \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} b +2 \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}}{\left (\sqrt {-a b}\, d +b c \right )^{2} \left (x -\frac {\sqrt {-a b}}{b}\right )}+\frac {3 \left (a \,d^{2}-2 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {2 b^{3} c^{6} \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3} \left (\sqrt {-a b}\, d +b c \right ) \left (\sqrt {-a b}\, d -b c \right )}+\frac {d^{3} a^{2} \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} b -2 \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}}{\left (\sqrt {-a b}\, d -b c \right )^{2} \left (x +\frac {\sqrt {-a b}}{b}\right )}-\frac {4 b^{4} c^{5} \left (3 a \,d^{2}+2 b \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{2} \left (\sqrt {-a b}\, d +b c \right )^{2} \left (\sqrt {-a b}\, d -b c \right )^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 b^{2} d^{3}}\)	\(648\)
default	\(\text {Expression too large to display}\)	\(1024\)

Input:

int(x^6/(d*x+c)^2/(b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*(-d*x+4*c)*(b*x^2+a)^(1/2)/b^2/d^3-1/2/b^2/d^3*(d^3*a^2/((-a*b)^(1/2) 
*d+b*c)^2/(x-(-a*b)^(1/2)/b)*((x-(-a*b)^(1/2)/b)^2*b+2*(-a*b)^(1/2)*(x-(-a 
*b)^(1/2)/b))^(1/2)+3*(a*d^2-2*b*c^2)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1 
/2)+2/d^3*b^3*c^6/((-a*b)^(1/2)*d+b*c)/((-a*b)^(1/2)*d-b*c)*(-1/(a*d^2+b*c 
^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b*c* 
d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d* 
(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b* 
c^2)/d^2)^(1/2))/(x+c/d)))+d^3*a^2/((-a*b)^(1/2)*d-b*c)^2/(x+(-a*b)^(1/2)/ 
b)*((x+(-a*b)^(1/2)/b)^2*b-2*(-a*b)^(1/2)*(x+(-a*b)^(1/2)/b))^(1/2)-4/d^2* 
b^4*c^5*(3*a*d^2+2*b*c^2)/((-a*b)^(1/2)*d+b*c)^2/((-a*b)^(1/2)*d-b*c)^2/(( 
a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+ 
b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/( 
x+c/d)))
 

Fricas [F(-1)]

Timed out. \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(x^6/(d*x+c)^2/(b*x^2+a)^(3/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^{6}}{\left (a + b x^{2}\right )^{\frac {3}{2}} \left (c + d x\right )^{2}}\, dx \] Input:

integrate(x**6/(d*x+c)**2/(b*x**2+a)**(3/2),x)
 

Output:

Integral(x**6/((a + b*x**2)**(3/2)*(c + d*x)**2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 581 vs. \(2 (227) = 454\).

Time = 0.12 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.32 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\frac {3 \, b^{2} c^{8} x}{\sqrt {b x^{2} + a} a b^{2} c^{4} d^{6} + 2 \, \sqrt {b x^{2} + a} a^{2} b c^{2} d^{8} + \sqrt {b x^{2} + a} a^{3} d^{10}} + \frac {3 \, b c^{7}}{\sqrt {b x^{2} + a} b^{2} c^{4} d^{5} + 2 \, \sqrt {b x^{2} + a} a b c^{2} d^{7} + \sqrt {b x^{2} + a} a^{2} d^{9}} - \frac {8 \, b c^{6} x}{\sqrt {b x^{2} + a} a b c^{2} d^{6} + \sqrt {b x^{2} + a} a^{2} d^{8}} - \frac {c^{6}}{\sqrt {b x^{2} + a} b c^{2} d^{6} x + \sqrt {b x^{2} + a} a d^{8} x + \sqrt {b x^{2} + a} b c^{3} d^{5} + \sqrt {b x^{2} + a} a c d^{7}} - \frac {6 \, c^{5}}{\sqrt {b x^{2} + a} b c^{2} d^{5} + \sqrt {b x^{2} + a} a d^{7}} + \frac {x^{3}}{2 \, \sqrt {b x^{2} + a} b d^{2}} - \frac {2 \, c x^{2}}{\sqrt {b x^{2} + a} b d^{3}} + \frac {5 \, c^{4} x}{\sqrt {b x^{2} + a} a d^{6}} - \frac {3 \, c^{2} x}{\sqrt {b x^{2} + a} b d^{4}} + \frac {3 \, a x}{2 \, \sqrt {b x^{2} + a} b^{2} d^{2}} + \frac {3 \, c^{2} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{b^{\frac {3}{2}} d^{4}} - \frac {3 \, a \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, b^{\frac {5}{2}} d^{2}} + \frac {3 \, b c^{7} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{9}} - \frac {6 \, c^{5} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{{\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{7}} + \frac {4 \, c^{3}}{\sqrt {b x^{2} + a} b d^{5}} - \frac {4 \, a c}{\sqrt {b x^{2} + a} b^{2} d^{3}} \] Input:

integrate(x^6/(d*x+c)^2/(b*x^2+a)^(3/2),x, algorithm="maxima")
 

Output:

3*b^2*c^8*x/(sqrt(b*x^2 + a)*a*b^2*c^4*d^6 + 2*sqrt(b*x^2 + a)*a^2*b*c^2*d 
^8 + sqrt(b*x^2 + a)*a^3*d^10) + 3*b*c^7/(sqrt(b*x^2 + a)*b^2*c^4*d^5 + 2* 
sqrt(b*x^2 + a)*a*b*c^2*d^7 + sqrt(b*x^2 + a)*a^2*d^9) - 8*b*c^6*x/(sqrt(b 
*x^2 + a)*a*b*c^2*d^6 + sqrt(b*x^2 + a)*a^2*d^8) - c^6/(sqrt(b*x^2 + a)*b* 
c^2*d^6*x + sqrt(b*x^2 + a)*a*d^8*x + sqrt(b*x^2 + a)*b*c^3*d^5 + sqrt(b*x 
^2 + a)*a*c*d^7) - 6*c^5/(sqrt(b*x^2 + a)*b*c^2*d^5 + sqrt(b*x^2 + a)*a*d^ 
7) + 1/2*x^3/(sqrt(b*x^2 + a)*b*d^2) - 2*c*x^2/(sqrt(b*x^2 + a)*b*d^3) + 5 
*c^4*x/(sqrt(b*x^2 + a)*a*d^6) - 3*c^2*x/(sqrt(b*x^2 + a)*b*d^4) + 3/2*a*x 
/(sqrt(b*x^2 + a)*b^2*d^2) + 3*c^2*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*d^4) - 
3/2*a*arcsinh(b*x/sqrt(a*b))/(b^(5/2)*d^2) + 3*b*c^7*arcsinh(b*c*x/(sqrt(a 
*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(5/2)*d 
^9) - 6*c^5*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d* 
x + c)))/((a + b*c^2/d^2)^(3/2)*d^7) + 4*c^3/(sqrt(b*x^2 + a)*b*d^5) - 4*a 
*c/(sqrt(b*x^2 + a)*b^2*d^3)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate(x^6/(d*x+c)^2/(b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx=\int \frac {x^6}{{\left (b\,x^2+a\right )}^{3/2}\,{\left (c+d\,x\right )}^2} \,d x \] Input:

int(x^6/((a + b*x^2)^(3/2)*(c + d*x)^2),x)
 

Output:

int(x^6/((a + b*x^2)^(3/2)*(c + d*x)^2), x)
 

Reduce [B] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 2568, normalized size of antiderivative = 10.27 \[ \int \frac {x^6}{(c+d x)^2 \left (a+b x^2\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int(x^6/(d*x+c)^2/(b*x^2+a)^(3/2),x)
 

Output:

(24*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
a*d + b*c*x)*a**2*b**3*c**6*d**2 + 24*sqrt(a*d**2 + b*c**2)*log( - sqrt(a 
+ b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**3*c**5*d**3*x + 12* 
sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
+ b*c*x)*a*b**4*c**8 + 12*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sq 
rt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**4*c**7*d*x + 24*sqrt(a*d**2 + b*c* 
*2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**4*c* 
*6*d**2*x**2 + 24*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d** 
2 + b*c**2) - a*d + b*c*x)*a*b**4*c**5*d**3*x**3 + 12*sqrt(a*d**2 + b*c**2 
)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**5*c**8*x 
**2 + 12*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c** 
2) - a*d + b*c*x)*b**5*c**7*d*x**3 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x) 
*a**2*b**3*c**6*d**2 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b**3*c** 
5*d**3*x - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**4*c**8 - 12*sqrt(a*d 
**2 + b*c**2)*log(c + d*x)*a*b**4*c**7*d*x - 24*sqrt(a*d**2 + b*c**2)*log( 
c + d*x)*a*b**4*c**6*d**2*x**2 - 24*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b 
**4*c**5*d**3*x**3 - 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**5*c**8*x**2 
- 12*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**5*c**7*d*x**3 - 16*sqrt(a + b*x 
**2)*a**4*b*c**2*d**7 - 10*sqrt(a + b*x**2)*a**4*b*c*d**8*x + 6*sqrt(a + b 
*x**2)*a**4*b*d**9*x**2 - 32*sqrt(a + b*x**2)*a**3*b**2*c**4*d**5 - 26*...